Cutting leaves triangular edges
You just deleted some central vertices, however if you want to cut you actually need to generate new vertices at the cutting plane (so no delete is necessary). To do this you can either:


Copy the mesh four times and do four boolean operations with some cubes you add (see boolean modifier)... this might be the easiest option
Or: Use the knife bisect tool in edit mode (and set the exact cutting plane in the popup menu after using knife bisect). You can then use seperate (select and press P) the parts as seperate objects.